Educational Display I designed but can't describe how it works

[industriald]

Hello all,

Apologies for asking a question as my first post.

I am helping out a friend by designing educational displays for an environmental centre. I do the concepts, CAD design, operate the CNC router and assemble the various components. I am unemployed and only just starting tertiary studies in my late 20's as I had a rough start to life, so apologies if this question is incredibly basic or obvious.

I did do some research but it is a bit hard to look for something when I don't exactly know what to look for.

Basically we have 3 parallel tracks that are different shapes, but they start at the same heights and end at the same heights. The 3 stainless steel ball bearings are released at the same time and run down the tracks and at the end hit a pointer which provides a relative angle which is designed to provide an indication of how much kinetic energy is remaining at the end of the run.

There is a total drop of 400mm vertically and a distance of 2100mm from start to finish horizontally. The Red track is 2410mm, the Blue is 2344mm and the Green is 2180mm.

The speeds of the run go from Red being the fastest (~2s), Blue being the next quickest(~3s) and Green being the slowest(~5s). We estimated that the Red track would be the fastest, followed by Green then Blue.

The thing that baffled us though is the angle indicated at the end by our pointer seems to be inverse to the speed - the Red deflects the least at 20 degrees, Blue being 35 degrees and Green being 40 degrees.

Can someone explain what is going on? We would be incredibly grateful to anyone that could help us out. I am trying to help a friend out and have built this marvelous contraption, but we can't really explain why it is working as it is. We have eliminated factors such as difference in the pointer mechanism by swapping them around and they all move freely.

We would be grateful if you could advise any formulas or laws we could look up, as we are not expecting it to be done for us! I think what we are looking for is conservation of momentum, but I am just not sure.

Thanks in advance

Ashley (industriald)

 

[AJ Bentley]

Beautiful piece of construction!
But you seem to have a problem with friction.

In theory, the kinetic energy at the end of the run should be the same for all paths.
(This is Conservation of Energy)
In fact, you seem to be losing energy based more-or-less on the length of run. The direct path is giving you the most efficient path.

I would suggest using the biggest, most massive balls you can find and try to take out any flexing or vibration in the track which needs to be a hard material.

 

[mfb]

Without friction, all balls would arrive at the same velocity.
Air resistance depends on speed - higher speed gives more friction. Therefore, the red ball (which is quicker at most of the track length) loses more energy than the others.

An experimental issue: Did you switch the tracks and/or re-adjust the pointers? They might react in different ways to the same impacts.

 

[willem2]

Without friction, all balls would arrive at the same velocity.

That wasn't the question. The problem is that how much time the balls take over the entire track.

The reason that the green track seems so slow is that the ball will travel very slowly in the first part, where the track is only a little below the starting point.
If the ball is at an altitude h below the strarting point, it will have lost an amount mgh of potential energy, which will be converted to kinetic energy (1/2 +1/5)mv^2
combining those you get

v = \sqrt{\frac {10}{7} g h}

because h is very small at the start for the green track, the velocity is very slow there, and there is no way to make up for that in the rest of the track.

(the term 1/5 mv^2 is from the rotation energy of a sphere)

 

[AJ Bentley]

That wasn't the question. The problem is that how much time the balls take over the entire track.

Read the question again. The OP is puzzled by the fact that the final KE scales oppositely to the average velocity. (It should of course be the same for each track.)

 

[sophiecentaur]

The KE differences could be because the red track gets the ball traveling faster than the others, earlier, so if the losses increase with speed, the red track will introduce more loss as it's going faster for a longer disgtance than the others. A track with the least loss would have a very small slope initially and a sharp drop at the end (the 'inverse' of the red one.) Time would be very great.

The average times would be expected in the order red (least) then probably blue then green. To find the average speed, you need a harmonic mean of speeds along the way on each elemental length of track.

So I think the results are largely what one should be expecting.

Lovely bit of kit, by the way!

 

[AlephZero]

You might want to add a model of the fastest possible track (ignoring friction, air resistance, etc) which is a cycloid. See http://en.wikipedia.org/wiki/Brachistochrone_curve

Then you could also start several balls at different points along the cycloid at the same time, and take bets on which finishes first: http://en.wikipedia.org/wiki/Tautochrone_curve

 

[industriald]

Hello guys,

Thank you all for the compliments and suggestions as to how to improve and explain it.

We really appreciate your replies.

The track itself is made from HDPE which is fairly hard and very slippery. The side walls are acrylic which is harder than the HDPE but a highly polished surface. I understand from the Big Bang Theory (The TV show) that friction can be reduced to a point so much that it introduces another source of friction. Could this be where the loses are coming from? This would explain why the stainless balls feel kind of "sticky" to the touch, even though they are not coated and are highly polished.

We eliminated sources of vibration and flexing as much as possible, it is supported at equal spacing along the length and also having the acrylic bolted to either side of it helps too.

There are 2 things that the experiment was meant to indicate - the velocity (the children can try to pick which one will finish first) and the kinetic energy available at the end.

As far as the pointers go, we did try and swap them around, and they all feel the same, but have no accurate way to measure it. I can't help but think it would be better with a bearing in the centre which would raise the pointer off the surface to further reduce friction. We had some skate bearings but the job requirement was stainless due to proximity to marine environments, so there is a possibility the bearings could have rusted without maintenance.

The other issue is that the track has small facets along it's length, instead of being a smooth curve. This is an problem with my friend's machine and the software used to generate the gcode (tool path). This seems fairly minor though and it makes a great noise when the balls are running. The facets are about 2mm long each.

sophiecentaur, your explanation makes sense to me. It is the exact opposite of what I assumed would happen, "the fastest track hits the hardest", but is exactly what is happening.

AlephZero, thanks for your suggestion regarding adding the fastest possible curve. Also we were looking to do a smaller display and the Tautochrone Curve would be perfect for that.

I have to say, that if the aim was to encourage discussion among the students and teachers using the device, then I think it has satisfied that requirement. I am so glad that my little creation has inspired such a vigorous discussion.

Cheers and thanks again for all your help.

Ashley (industriald)

 

[sophiecentaur]

It struck me that you could 'estimate' how much actual loss there is, due to friction. If you could arrange for a short curved slope from the standard height to shoot a ball at your pointer, that would give you an idea of how much KE would be available in the absence of friction along all that length of track. If this produces only slightly less than the best result so far, then the friction is not too much. (Arm waving, I know but it's still extra information about the setup)

 

[AJ Bentley]

it makes a great noise when the balls are running.

Oh dear! Noise = sound energy.

Noise in any machine is a BAD thing. Would you want your car to sound like that?

 

[mfb]

Noise = sound energy.

I would expect that the total energy going to sound is small.

The balls have a mass of something like ~10g (?), with 40cm height difference this corresponds to a total energy of 0.04J. To get a significant effect within 2-4s and 10% time with sound, the sound would need a power of at least ~10mW, similar to a helicopter. Unlikely.

 

[ImaLooser]

The KE differences could be because the red track gets the ball traveling faster than the others, earlier, so if the losses increase with speed, the red track will introduce more loss as it's going faster for a longer disgtance than the others. A track with the least loss would have a very small slope initially and a sharp drop at the end (the 'inverse' of the red one.) Time would be very great.

The average times would be expected in the order red (least) then probably blue then green. To find the average speed, you need a harmonic mean of speeds along the way on each elemental length of track.

So I think the results are largely what one should be expecting.

Lovely bit of kit, by the way!

Right. I would think that drag from the air would be significant. A spinning sphere is an especially unaerodynamic shape, and the drag increases exponentially with the speed. You could reduce the drag by using golf balls or something else with a dimpled surface, but that complicates things even more.

I think you won't be able to remove these effects. It might be better to explain them. "The higher the speed, the greater the loss." That's worth knowing.

 

[AJ Bentley]

I would expect that the total energy going to sound is small.

The balls have a mass of something like ~10g (?), with 40cm height difference this corresponds to a total energy of 0.04J. To get a significant effect within 2-4s and 10% time with sound, the sound would need a power of at least ~10mW, similar to a helicopter. Unlikely.

I disagree. Noise is only a symptom of wasted energy really. Most of the actual wasted energy will dissipate by the usual mechanisms. But it means that the balls are not rolling smoothly, they are bouncing from facet to facet.

Every bounce and return could lose an amount of energy in the order of the height of the leap. (I doubt that the track is rigid enough to provide a very high co-efficient of restitution)

If the facets are 2mm, each ball bounces around 1000 times along the length of the track. That means it only needs to bounce 0.04mm on each facet to lose energy equivalent to 10% of 40cm.
Even allowing for a restitution of 50-60% it's going to be significant.

This sort of inclined-plane mechanism is usually quite efficient. Something is different about this one.

P.S. Are those balls really only 10g or is that a (bad) guess? That sounds far too small and light to hang onto it's KE very well. combined with the facets I'm surprise any of them make it to the end! (Checking, the OP doesn't mention ball weight)

 

[mfb]

I disagree. Noise is only a symptom of wasted energy really. Most of the actual wasted energy will dissipate by the usual mechanisms. But it means that the balls are not rolling smoothly, they are bouncing from facet to facet.

I agree, and did not write anything else in my post.
Noise -> negligible sound energy (, but other losses).

P.S. Are those balls really only 10g or is that a (bad) guess? That sounds far too small and light to hang onto it's KE very well. combined with the facets I'm surprise any of them make it to the end! (Checking, the OP doesn't mention ball weight)

That was just a guess, based on the image (note the small holes there) and the approximate density of steel.

 

[sophiecentaur]

Do you have any datalogging equipment? You could measure speeds at different parts of the track. That would make calculating losses very easy.